
Simple Genetic Algorithm (\({\mbox{\texttt {GA}}}\) ):
to evolve local population, \({\mbox{\texttt {GA}}}\)  considers a breeding cycle where the best parents are selected and produce offspring by
applying one-point crossover (Fig. REF  (a)). Then, the descendants replace the worst individuals in the current population. The breeding cycle relies on four parameters, namely, the percentages of selection and replacement, and the probability of application of mutation and crossover. \({\mbox{\texttt {GA}}}\)  was developed by J. H. Holland in the 1970s [1]}.

Double-point Crossover Genetic Algorithm (\({\mbox{\texttt {GAD}}}\) ):
it has a similar behavior than \({\mbox{\texttt {GA}}}\)   except by the technique to promote crossover, illustrated in Fig. REF  (b), and how the local population evolves: in contrast with \({\mbox{\texttt {GA}}}\)   descendants replace individuals randomly selected in \({\mbox{\texttt {GAD}}}\) .

Differential Evolution (\({\mbox{\texttt {DE}}}\) ):
it was proposed by Storn and Price [2]}, and is a method to optimize functions over the real multidimensional space \(\mathbb {R}^n\) . We adapt the algorithm by restricting the domain of the function as the set of permutations. Two main parameters guide the evolutionary process: the mutation factor \(F_M\) , applied to individuals randomly selected from the population to generate mutants, and the probability of crossover \(P_C\) .
The local population evolves by replacing individuals having the worst fitness with mutants.

Self-adjusting Particle Swarm Optimization (\({\mbox{\texttt {PSO}}}\) ):
it was introduced by Eberhart and Kennedy [3]} and is based on the behavior of social organisms in groups. Originally, \({\mbox{\texttt {PSO}}}\)  was developed to optimize continuous functions from particles' velocity and position in an \(n\) -dimensional space. At each iteration, the vector representing a particle velocity is built from the best positions of such a particle and all particles, the velocity of the particle in the previous iteration, the individual and global acceleration (that influence the distance a particle can cover), and the weight of inertia (momentum). In this paper, we use the \({\mbox{\texttt {PSO}}}\)  proposed in [4]}, which is self-adaptive since momentum and the individual and global acceleration coefficients are self-tuning during the search process.

